nLab
Lie derivative

Idea

Given a smooth manifold $M$ and a vector field $X\in \Gamma (TM)$ on it, one defines a series of operators ${ℒ}_X$ on spaces of differential forms, of functions, of vector fields and multivector fields. For functions ${ℒ}_X(f)=X(f)$ (derivative of $f$ along an integral curve of $X$); as multivector fields and forms can not be compared in different points, one pullbacks or pushforwards them to be able to take a derivative.

For vector fields ${ℒ}_{X}Y=[X,Y]$. If $\omega \in {\Omega }^{•}(M)$ is a differential form on $M$, the Lie derivative ${ℒ}_X\omega$ of $\omega$ along $X$ is the linearization of the pullback of $\omega$ along the flow $\exp (X-):ℝ\times M\to M$ induced by $X$

$${ℒ}_X\omega =\frac{d}{dt}{\mid }_{t=0}\exp (tX{)}^{*}(\omega ).$$\mathcal{L}_X \omega = \frac{d}{d t}|_{t = 0} \exp(t X)^*(\omega) \,.

Denote by ${\iota }_X:{\Omega }^{•}(M)\to {\Omega }^{•-1}(M)$ be the graded derivation which is the contraction with a vector field $X$. By Cartan's homotopy formula,

$${ℒ}_v=[d_{\mathrm{dR}},{\iota }_v]=d_{\mathrm{dR}}\circ {\iota }_v+{\iota }_v\circ d_{\mathrm{dR}}:{\Omega }^{•}(X)\to {\Omega }^{•}(X).$$\mathcal{L}_v = [d_{dR}, \iota_v] = d_{dR} \circ \iota_v + \iota_v \circ d_{dR} : \Omega^\bullet(X) \to \Omega^\bullet(X) \,.

References

An introduction in the context of synthetic differential geometry is in

• Gonzalo Reyes, Lie derivatives, Lie brackets and vector fields over curves, pdf

A gentle elementary introduction for mathematical physicists

• Bernard F. Schutz, Geometrical methods of mathematical physics (elementary intro) amazon, google

.